Overcategory

In mathematics, specifically , an overcategory (and undercategory) is a distinguished class of used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object in some category . There is a dual notion of undercategory, which is defined similarly. Let be a category and a fixed object of pg 59. The overcategory (also called a slice category) is an associated category whose objects are pairs where is a morphism in . Then, a morphism between objects is given by a morphism in the category such that the following diagram commutesThere is a dual notion called the undercategory (also called a coslice category) whose objects are pairs where is a morphism in . Then, morphisms in are given by morphisms in such that the following diagram commutesThese two notions have generalizations in and pg 43, with definitions either analogous or essentially the same. Many categorical properties of are inherited by the associated over and undercategories for an object . For example, if has finite and coproducts, it is immediate the categories and have these properties since the product and coproduct can be constructed in , and through universal properties, there exists a unique morphism either to or from . In addition, this applies to and colimits as well. Recall that a site is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category whose objects are open subsets of some topological space , and the morphisms are given by inclusion maps. Then, for a fixed open subset , the overcategory is canonically equivalent to the category for the induced topology on . This is because every object in is an open subset contained in . The category of commutative -algebras is equivalent to the undercategory for the category of commutative rings. This is because the structure of an -algebra on a commutative ring is directly encoded by a ring morphism .